# prove that τ is the topology on ℝ

#### blbl

Define τ⊂ P(ℝ) as follows:
τ={φ}∪{u⊆ℝ:{-π,π}⊆u} .
prove that τ is a topology on ℝ .

Last edited:

#### Drexel28

MHF Hall of Honor
Define τ⊂ P(ℝ) as follows:
τ={φ}∪{u⊆ℝ:{-π,π}⊆u} .
prove that τ is the topology on ℝ .
Is this $$\displaystyle T=\{\varnothing\}\cup\left\{U\subseteq\mathbb{R}-\pi,\pi)\subseteq U\right\}$$? What's wrong? clearly $$\displaystyle \varnothing,\mathbb{R}\in T$$ if $$\displaystyle (-\pi,\pi)\subseteq U_\alpha$$ then $$\displaystyle (-\pi,\pi)\subseteq\bigcup_{\alpha\in\mathcal{A}}U_\alpha$$ and similarly for the intersection. Or, is this $$\displaystyle T=\{\varnothing\}\cup\left\{U\subseteq\mathbb{R}:\{-\pi,\pi\}\subseteq U\right\}$$? It's the same.